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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Injective modules and localization in noncommutative Noetherian rings


Author: Arun Vinayak Jategaonkar
Journal: Trans. Amer. Math. Soc. 190 (1974), 109-123
MSC: Primary 16A08
DOI: https://doi.org/10.1090/S0002-9947-1974-0349727-X
MathSciNet review: 0349727
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Abstract: Let $ \mathfrak{S}$ be a semiprime ideal in a right Noetherian ring R and $ \mathcal{C}(\mathfrak{S}) = \{ c \in R\vert[c + \mathfrak{S}$ regular in $ R/\mathfrak{S}\} $. We investigate the following two conditions:

$ ({\text{A}})\;\mathcal{C}(\mathfrak{S})$ is a right Ore set in R.

$ ({\text{B}})\;\mathcal{C}(\mathfrak{S})$ is a right Ore set in R and the right ideals of $ {R_{\mathfrak{S}}}$, the classical right quotient ring of R w.r.t. $ \mathcal{C}(\mathfrak{S})$ are closed in the $ J({R_{\mathfrak{S}}})$-adic topology.

The main results show that conditions (A) and (B) can be characterized in terms of the injective hull of the right R-module $ R/\mathfrak{S}$. The J-adic completion of a semilocal right Noetherian ring is also considered.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0349727-X
Keywords: Right Noetherian ring, localization at a semiprime ideal, classical localization, Ore set, AR-property, completion, injective modules
Article copyright: © Copyright 1974 American Mathematical Society